For my laptop, to be able to enroll in the Extended Security Updates, first I had to remove my "Work or School Account" from Settings -> Accounts -> Access Work or School. I clicked on the Work account that was there, clicked "Disconnect", and restarted the computer. After the restart, I was able to go to Settings -> Update and Security and enroll using the link that directed me to the Microsoft Website explainer on "End of Support for Windows 10". Scroll down the page to the "Extended Security Updates" link and follow the steps.
Phew. It took several hours of trying everything else and not getting anywhere, then finally the above worked. I guess Windows thinks my computer is ineligible if I have a Work or School Account associated with my machine.
Apparently disconnecting the Work/School account does not delete the account itself, it just removes the sign-in information. Therefore, it should be possible to add it back again. I will be able to find that out once the weekend is over and my work's IT department is open. Fingers crossed.